Development of Attribute Preserving Network Equivalents
Thomas J. Overbye ([email protected]) Fox Family Professor of Electrical and Computer Eng.
University of Illinois at Urbana-Champaign
Graduate students Wonhyeok Jang ([email protected])
Saurav Mohapatra ([email protected])
1
CERTS Meeting Aug. 5-6, 2014
Overview Overall objective
– To develop equivalent systems that preserve desired attributes of the original system
Present focus – To create equivalents of interconnection level power
grids that preserve line limits Special emphasis
– To apply algorithm to a “backbone” type equivalent of large systems such as Eastern Interconnection (EI)
– Key result is limits have been assigned to the EI equivalent provided by the Tylavsky group
2
Background For decades, power system network models
have been equivalenced using the approach originally presented by J. B. Ward in 1949 AIEE paper “Equivalent Circuits for Power-Flow Studies”
3
Studysystem
External
system
Boundarybuses
Studysystem
Boundarybuses
Ward Equivalents Gaussian elimination on nodal admittance matrix
– Also known as Kron reduction Admittance matrix updated by eliminating one
bus at a time with partial factorization – More efficient than inverting admittance matrix – When bus k between bus i and bus j is equivalenced
– Equivalent line limits are missing
4
ijijkk
kjikijij yy
yyy
yy' ~−=−=i
j k
i
j
ijyijyiky
jkyijy~
Summary of Accomplishments Developed algorithms for calculating equivalent line
limits by matching total transfer capacity (TTC) – Prototype of algorithm (Max/Hungarian): W. Jang, S. Mohapatra,
T. J. Overbye and H. Zhu, “Line Limit Preserving Power System Equivalent,” in Proc. 2013 PECI, Feb. 2013.
– Improved algorithm (Quadratic Program): S. Mohapatra, W. Jang and T.J. Overbye, “Equivalent Line Limit Calculation for Power System Equivalent Networks,” IEEE Trans. Power Systems, available as an Early Access Article.
– Modified algorithm for interconnection-level systems (Top-down approach): W. Jang, S. Mohapatra, and T. J. Overbye, “Towards a Transmission Line Limit Preserving Algorithm for Large-scale Power System Equivalents,” submitted to HICSS 2015.
5
Previous Algorithms Used a sequential bus elimination approach,
updating limits as buses were eliminated – Used Power Transfer Distribution Factors (PTDFs) and
Total Transfer capability (TTCs) to calculate the limits – Even simple, unloaded systems could have no exact
solution – required a range for the limits – Various algorithms were considered to determine the
maximum upper/lower bounds for limits • Assignment problem approach (Hungarian algorithm) could
result in wide limit ranges • Quadratic approach resulted in a smaller range, but could be
quite expensive computationally
6
Previous Algorithm 118 Bus Case 7
15
14
13
12
117
16
1
2
3
4
11
5 6
7
8
9
17
30
113
31
29
28
27
32
114 115
18 19
33
34
37
20
21
22
23
25
26
35
36
38
39
40 41 42
43
44
45
46 47
48
49
52
5354
50 51
57 58
56
55
60
59
61
62
64
63
65
66
67
6869
24
72
70
71
73
74
75 118
76
77
78
116
79
81
80
82
83
84
85
86
97
96
98
88
95 94
89 92
90
91
93
99
102 101
100
103
104
105
106
107
108
109
110
111 112
87
0. 97 pu
0. 97 pu0. 97 pu
0. 97 pu
0. 97 pu
0. 97 pu 0. 97 pu
0. 97 pu
0. 97 pu0. 95 pu
0. 98 pu
0. 95 pu
0. 98 pu
0. 98 pu
0. 97 pu
0. 95 pu
0. 99 pu0. 99 pu
0. 99 pu
0. 95 pu
0. 99 pu 0. 99 pu
0. 99 pu
0. 99 pu 0. 99 pu
0. 95 pu
0. 99 pu
0. 99 pu
1. 00 pu
1. 00 pu
0. 99 pu
0. 99 pu
0. 99 pu 0. 99 pu
1. 01 pu
0. 99 pu
1. 00 pu
1. 01 pu
1. 01 pu
1. 01 pu 0. 95 pu
1. 01 pu
1. 00 pu 1. 02 pu
1. 01 pu
1. 02 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 02 pu
1. 02 pu
1. 02 pu
1. 02 pu
1. 01 pu
1. 02 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
0. 97 pu
0. 97 pu
0. 96 pu
1. 02 pu
1. 02 pu
1. 01 pu
1. 02 pu1. 01 pu1. 01 pu
1. 01 pu 1. 01 pu
1. 01 pu
1. 01 pu
1. 02 pu
1. 02 pu
1. 02 pu
1. 01 pu
1. 03 pu
1. 03 pu
0. 96 pu
0. 96 pu
1. 03 pu 1. 03 pu 1. 03 pu
1. 03 pu
1. 03 pu 1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu 1. 03 pu 1. 03 pu
1. 03 pu
1. 00 pu
0. 97 pu
1. 00 pu
1. 00 pu
0. 97 pu
1. 01 pu
1. 01 pu
1. 03 pu1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
1. 03 pu
Previous Algorithms 118 Bus Case Reduced to 30 Buses
8
15
14
13
12
16
3
4
11
5 6
7
8
9
17
30
113
31
29
28
27
32
114 115
18 19
33
34
37
20
21
22
23
25
26
35
36
38
39
43
44
45
46 47
48
49
52
53
50 51
57 58
60
59
61
62
64
63
65
66
67
6869
24
72
70
71
73
74
75 118
76
77
78
116
79
81
80
82
83
84
85
86
97
96
98
88
95 94
89 92
90
91
93
99
102 101
100
103
104
105
110
0. 05 pu
0. 04 pu
0. 04 pu
0. 04 pu
0. 04 pu
0. 01 pu
0. 00 pu
0. 06 pu
0. 30 pu
0. 07 pu
0. 00 pu
0. 01 pu
0. 01 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0. 00 pu
0 M W 0 M var
Black lines represent fully retained lines between buses from the original case. Green lines correspond to equivalent lines, now with limits
Alternative, Top-down Approach Modified algorithm for large scale systems
– Still based on TTC matching using PTDFs Creates equivalents from the system-level and
proceeds downwards – Previous algorithm worked sequentially at the bus-level
(bottom-up approach) Can handle loaded network Consists of two main parts
– Creation of the equivalent without equivalent line limits – Assignment of equivalent line limits
9
Creation of Equivalent Goal
– To create a backbone type equivalent without breaking up generation and possibly load injections
Method – As with any equivalent, study/external buses must be
selected a priori – Explicitly retained external generators and loads are
assigned to an internal bus; this defines a group Criterion for grouping buses
– Substations have been used in this work – Other criteria can also be used (application dependent)
10
Procedure for Creating Equivalent
Select buses to be retained in the study system Assign each external bus to a study bus's group For each group
– Move all generators, loads, and shunts from external buses to study buses; loads and/or generators can be split if desired
– Combine any generators, load and/or shunts with similar types if desired
– Move DC lines from external buses to study buses in both groups
– Eliminate the external buses using Kron reduction
– Discard equivalent lines above a desired impedance threshold
11
Bus Grouping 12
Full system converted to equivalent system
A
C
GroupAlpha
GroupCharlie
B
GroupBravo
W
X
YZ
V
A
C
B
W
X
YZ
V
Study busExternal bus
Retained line
Eliminated lineEquivalent line Group
Note: Only equivalent lines that are below the
desired threshold are kept
4 Bus Example Assume buses 1 and 3 are a
group with bus 1 the only external bus
When removing bus 1 – Its generator is moved to bus 3 – Bus 1 is eliminated using Kron
reduction – Three equivalent lines are
created between the other buses
13
j0.12j0.14
j0.06
j0.10j0.08
slack
1
2
3
4
80 MVA 90 MVA
100 MVA
60 MVA 70 MVA
138.00 kV
137.71 kV
138.00 kV
229.91 kV
100 MW 0 Mvar
100 MW 0 Mvar
j0.27
j0.54
j0.23
j0.10j0.08
slack
2
3
4
80 MVA 90 MVA
138.00 kV
137.71 kV
230.00 kV
100 MW 0 Mvar
100 MW 0 Mvar
Criteria for Moving Generators and Loads to Study Buses
During the Kron reduction process the equations solved are
How the generation and load should get moved is determined by the left-hand side vector – This allocates the injection among the study buses
14
This is done efficiently with a partial factorization of the Ybus
Criteria for Moving Generators and Loads to Study Buses
One approach for discretely moving a generator to a study bus is to calculate how the generator's current injection would get allocated to the study buses, and then pick the one with the highest allocation – These allocations can be solved quite efficiently
during the Ybus partial factorization – Would only need to be done for the largest generators – Allocations are usually quite localized, with significant
values at just a handful of study buses
15
Eastern Interconnect Example Image shows percentage values of the UIUC
generation to a 300+kV backbone equivalent
16
UIUC 138 kV Gen Remvoed
53.3%20.9%
6.2%
5.6%
5.7%
Sul l i vanAEP765. 00 kV
BreedAEP
345. 00 kV
CaseyAMIL
345. 00 kV
D arwi nAEP
345. 00 kV
EugeneAEP
345. 00 kV
N ucorD EM345. 00 kV
1MaroaAMIL
345. 00 kV
D equi neAEP
345. 00 kV
345 00
If generator is to be left whole, then it should be moved to the 53.3% substation (Sidney)
Assignment of Equivalent Line Limits
Criteria for line limit preservation – Matching TTCs between pairs of buses for all
equivalent lines with those of the same pairs in the original system (but source/sink may be distributed)
TTC for transaction w between bus x and bus y, 𝑃𝑃𝑤𝑤(𝑥𝑥,𝑦𝑦) can be calculated as
– 𝐹𝐹𝑙𝑙𝑖𝑖: limit of line 𝑙𝑙𝑖𝑖 from the set of eliminated lines L
– 𝜑𝜑𝑙𝑙𝑖𝑖𝑤𝑤(𝑥𝑥,𝑦𝑦): PTDF on line 𝑙𝑙𝑖𝑖 for the same transaction w
17
=∈ y)w(x,
l
l
l
y)w(x,
i
i
i φF
PL
min
Four Bus Example (Exact solution case)
PTDFs shown for bus 2-3 When removing bus 1
𝑃𝑃𝑤𝑤(2,3) = min70
0.32,
1000.26
,60
0.06
= 217.0 MW (1-3 binding) Likewise, 𝑃𝑃𝑤𝑤(2,4) = 171.7 MW (1-4 binding) 𝑃𝑃𝑤𝑤(3,4) = 145.0 MW (1-4 binding)
18
j0.12 j0.14
j0.06
j0.10j0.08
slack
1
2
3
4
26%PTDF
32%PTDF
6%PTDF
68%PTDF
6%PTDF
80 MVA 90 MVA
100 MVA
60 MVA 70 MVA
138.00 kV
137.71 kV
138.00 kV
229.91 kV
100 MW 0 Mvar
100 MW 0 Mvar
Assignment of Equivalent Line Limits
For the original system calculate the TTC for the transaction between buses x and y, 𝑃𝑃𝑤𝑤(𝑥𝑥,𝑦𝑦) , only considering limits on lines to be eliminated – Only the study buses are included in this calculation – Distributed source and sink injections can be used
Using the equivalent system, calculate the PTDF on equivalent line x-y for transaction bus x to bus y, 𝜑𝜑�𝑙𝑙(𝑥𝑥,𝑦𝑦)𝑤𝑤(𝑥𝑥,𝑦𝑦)
Assign limit of equivalent line x-y as
19
y)w(x,y)(x,l
y)w(x,y)(x,l φPF ~~
~~ ×=
j0.27
j0.54
j0.23
j0.10j0.08
slack
2
3
4
80 MVA 90 MVA
137.45 kV
137.21 kV
230.00 kV
100 MW 0 Mvar
100 MW 11 Mvar
23%PTDF
2%PTDF
9%PTDF
68%PTDF
6%PTDF
Four Bus Example (Exact solution case)
PTDFs shown for bus 2-3 Equivalent line limit 2-3 is
𝐹𝐹�𝑙𝑙(2,3)
= 𝑃𝑃𝑤𝑤(2,3) × 𝜑𝜑�𝑙𝑙(2,3)𝑤𝑤(2,3)
= 217.0 X 0.23 = 50.8 MVA Likewise, 𝐹𝐹�𝑙𝑙(2,4)
= 171.7 X 0.24 = 41.4 MVA
𝐹𝐹�𝑙𝑙(3,4) = 145.0 X 0.20 = 28.5 MVA
20
Max/Hungarian Quadratic program
𝐹𝐹�𝑙𝑙(2,3) 50.8 MVA 50.8 MVA
𝐹𝐹�𝑙𝑙(2,4) 41.4 MVA 41.4 MVA
𝐹𝐹�𝑙𝑙(3,4) 28.5 MVA 28.5 MVA
Four Bus Example (Non-exact solution case)
Limit 1-4 reduced to 20 MVA from 60 MVA
21
j0.12 j0.14
j0.06
j0.10j0.08
slack
1
2
3
4
27%PTDF
8%PTDF
35%PTDF
8%PTDF
65%PTDF
80 MVA 90 MVA
100 MVA
20 MVA 70 MVA
138.00 kV
137.71 kV
138.00 kV
229.91 kV
100 MW 0 Mvar
100 MW 0 Mvar
Top-down
Max/ Hungarian Quadratic Program
Upper Lower Upper Best Lower
𝐹𝐹�𝑙𝑙(2,3) 50.8 50.8 50.8 50.8 50.8 50.8
𝐹𝐹�𝑙𝑙(2,4) 13.8 13.8 13.8 13.8 11.7 10.5
𝐹𝐹�𝑙𝑙(3,4) 9.5 19.2 9.5 19.2 19.2 19.2
mis-match* 29.1% 18.2% 29.1% 18.2% 11.0% 13.9%
*rms normalized TTC mismatch New algorithm provides a lower bound on the limits
Application to Eastern Interconnect (EI) Case
The next several slides provide examples of the application of the top-down algorithm applied to a 2012 EI model – Case originally had about
62,000 buses – Simulations were done using
SimAuto in PowerWorld and Matlab code
– Had to deal sometimes with case "quirks"
22
Negative Reactance Lines Testing indicates less likely to have exact
solutions in networks with negative reactance (capacitive) lines – Negative reactances can cause PTDFs to have values
above 100% – Negative reactances can occur on branches in series
compenstated lines, but the net line reactance is positive
– EI case has about 1400 branches with X < 0; usually arise because of the modeling of three-winding transformers
23
Negative Reactance Lines, Four Bus Example
Example shows previous four bus case with the reactance on the line from 2-3 changed from 0.08 to negative 0.08 – PTDFs calculated for
a transfer from bus 2 to bus 3
24
j0.12 j0.14
j0.06
j0.10-j0.08
slack
1
2
3
4
73%PTDF
91%PTDF
18%PTDF
191%PTDF
18%PTDF
80 MVA 90 MVA
100 MVA
20 MVA 70 MVA
137.90 kV
137.70 kV
138.00 kV
229.59 kV
100 MW 0 Mvar
100 MW 7 Mvar
Three Winding Transformer Equivalent Circuit
25
Image Source: http://w3.usa.siemens.com/datapool/us/SmartGrid/docs/pti/2010July/PDFS/Modeling%20of%20Three%20Winding%20Voltage%20Regulating%20Transformers.pdf
AEP and LGEE use 999 kV as nominal voltage for the star bus, others use 1 kV, some use the kV for one of the windings
Working with the Tylavsky Group EI Model
A desired outcome from the project is to provide a meaningful limits for the equivalent lines in the EI case being developed by Tylavsky group
A success of project is limits have been assigned, using the top-down approach, to the equivalent lines in this model – Since an equivalent case was provided, there is no
need to create the equivalent – Prevent algorithm provides lower bound limits – A next step is determine the impact of these limits
26
Summary of EI Equivalent
27
Original case Equivalent case 𝑬𝑬𝑬𝑬𝑬𝑬.𝑶𝑶𝑶𝑶𝑶𝑶.
× 𝟏𝟏𝟏𝟏𝟏𝟏 (%)
Number of buses 62,013 5,222 8.4
Number of branches 76,536
14,092 (retained: 6,694,
equivalent: 7,398) 18.4
Number of branches with negative
reactance
1378 (average: -0.028 pu)
87 (average: -0.314 pu) 6.3
Number of branches with infinite limits 25,186 112 0.4
Number of overloaded
original branches 0 61 n/a
Average % of original branch overloading 0 546 n/a
Generation 664,850 MW 664,691 MW 100.0 Load 647,898 MW 664,691MW 102.6
Analysis of Equivalent (TTC matching)
Table shows a comparison of study bus transactions between the original and the reduced system
28
Transaction Number of buses in between
𝑇𝑇𝑇𝑇𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜 (MW) 𝑇𝑇𝑇𝑇𝑇𝑇𝑒𝑒𝑒𝑒 (MW) 𝑻𝑻𝑻𝑻𝑻𝑻𝒐𝒐𝑶𝑶𝑶𝑶−𝑻𝑻𝑻𝑻𝑻𝑻𝒆𝒆𝑬𝑬
𝑻𝑻𝑻𝑻𝑻𝑻 𝒐𝒐𝑶𝑶𝑶𝑶×
𝟏𝟏𝟏𝟏𝟏𝟏 (%)
A (500 kV - 500 kV) 0 1133.5 1099.5 3.0
B (765 kV – 765 kV) 0 4422.8 4385.1 0.9
C (765 kV – 345 kV) 0 4082.0 1891.4 53.7
D (500 kV – 345 kV) 0 2527.8 2524.4 0.1
Analysis of Equivalent (TTC matching)
Transaction C also has negative reactance lines TTC mismatch is greatly reduced when there is
no negative reactance lines involved in calculating the TTCs
29
Transaction Number of buses in between
𝑇𝑇𝑇𝑇𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜 (MW) 𝑇𝑇𝑇𝑇𝑇𝑇𝑒𝑒𝑒𝑒 (MW) 𝑻𝑻𝑻𝑻𝑻𝑻𝒐𝒐𝑶𝑶𝑶𝑶−𝑻𝑻𝑻𝑻𝑻𝑻𝒆𝒆𝑬𝑬
𝑻𝑻𝑻𝑻𝑻𝑻 𝒐𝒐𝑶𝑶𝑶𝑶×
𝟏𝟏𝟏𝟏𝟏𝟏 (%)
E (500 kV – 230 kV) 4 826.3 829.0 0.3
F (500 kV – 230 kV) 4 2173.4 2061.4 5.2
G (500 kV – 500 kV) 4 9585.6 9594.0 0.1
H (345 kV – 230 kV) 3 471.7 464.2 1.6
Comments on Equivalent Line Limit Calculations
30
PTDF threshold in TTC calculation – Lines with low PTDF would unlikely be the binding
constraint, especially in backbone-type equivalent – With a 5% threshold, the number of lines to consider
in the TTC calculation is reduced from total lines in case to an average of just dozens of lines
Zero MVA line limits in the original case indicate no limit is enforced – Values replaced with either 9,999 MVA or they can be
capped with a maximum power transfer value 𝑉𝑉1𝑉𝑉2𝑋𝑋 sin𝜃𝜃
Computational Aspects Problem formulation is straightforward compared
to Quadratic program Computation is linear with respect to the number
of equivalent lines – Applicable to large scale systems – In contrast, computation in bottom-up algorithms increased
exponentially with respect to the number of first neighbor buses of eliminated buses/groups
Two factors for control of computation vs accuracy – Impedance threshold to discard high impedance lines – PTDF threshold in calculating TTCs
31
Future Work Increase accuracy of the results
– Develop algorithm to provide an upper bound on limits, in order to reduce large TTC mismatches
– Dealing with lines with negative reactance
Work with the Tyavsky group to determine impact of the limits on the solution results
32